Projective real calculi and Levi-Civita connections

Based on its central role in the framework of real calculi, the existence of the Levi-Civita connection for real calculi over projective modules is studied, with a special emphasis placed on the simple module of N-dimensional complex vectors over the algebra of complex N-by-N matrices. It is demonstrated that existence of the Levi-Civita connection in this case depends on the Lie algebra g of hermitian derivations, and necessary and sufficient conditions for the possibility of constructing a real calculus for which there exists a Levi-Civita connection are given in this case. Furthermore, in the general case of real calculi over projective modules, necessary and sufficient conditions for the existence of the Levi-Civita connection are given in terms of explicit projection coefficients.